#203775

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,32)(31,35)(33,36);(1,3,4,2)(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)(21,23,25,32,26,33,24,31)(22,27,28,30,29,35,34,36);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,d^c=dh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110971);